Morse indices of Yang-Mills connections over the unit sphere

C ^OMPOSITIO M ^ATHEMATICA

S ^HIN N ^AYATANI H ^AJIME U ^RAKAWA

Morse indices of Yang-Mills connections over the unit sphere

Compositio Mathematica, tome 98, n

^o

2 (1995), p. 177-192

<http://www.numdam.org/item?id=CM_1995__98_2_177_0>

© Foundation Compositio Mathematica, 1995, tous droits réservés.

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Morse indices of Yang-Mills connections over the unit sphere

Dedicated ^to

Professor

^{Masaru Takeuchi on} his sixtieth

birthday

SHIN

NAYATANI1

and HAJIME

URAKAWA2

1 Mathematical Institute, ^Tohoku University, Aoba, Sendai, 980, Japan

2Mathematics Laboratories, Graduate School of Information Sciences, ^Tohoku University, Katahira, Sendai, 980, Japan

Received 19 October 1993; accepted in final form 27 May ¹⁹⁹⁴

Abstract. We give ^an optimal estimate for the (Morse) ^index

i(0)

of any nonflat Yang-Mills

connection V over the m-sphere ^sm

(m 5)

with the standard Riemannian metric as follows:

i(~)

^{m + 1.}

Notice the canonical connection on S’n ⁼

SO(m

⁺

1)/SO(m) (m 5)

achieves the equality.

© 1995 Kluwer Academic Publishers. Printed in the Netherlands.

1. Introduction

We consider a

compact

Riemannian manifold

M,

^a

principal

^{G-bundle P over}

M,

and the associated G-vector bundle E over

M,

^{where G is a}

compact

Lie group. On the space

C ( E )

^of G-connections of

E,

^{we consider the}

Yang-Mills functional:

where

Rv

is the curvature of the connection ~ and the norm is defined in terms of the Riemannian

metric g

^{on M and a fixed}

Ad(G)-invariant

^inner

product

^{on the}

Lie

algebra

g of G.

A critical

point

^of

^y.M

is called a

Yang-Mills

connection. Our interests are

in the second variation of the functional at such a connection and

instability

^of

Yang-Mills

connections. A

Yang-Mills

connection is said to be

weakly

^stable

if the second variation

of y M

^{at V is}

non-negative, i.e.,

for every smooth

one-parameter family Vt, Itl

^{E, with}

^{~0 =}

^V.

Otherwise,

we say that V is unstable.

Furthermore, Bourguignon

and Lawson

[B.L]

gave

the second variation formula and the notions of

(Morse)

^index

i(0)

^and

nullity

n(~) of a Yang-Mills

connection B7.

Roughly speaking,

^the index is the dimension of the space of infinitesimal deformations

decreasing y M

^{and the}

nullity

^{is the}

dimension of the space of those

preserving ^YM (cf.

Definition

2.5).

^Thus

instability

of a

Yang-Mills

connection i7 means

i(~)

¹

^(cf.

Definition

2.6).

^’

At the

Tokyo Symposium

^on "Minimal Submanifolds and Geodesics" in

Septem-

ber of

1977,

J. Simons announced the

following

^{theorem in his talk:}

THEOREM ^{1.3. For} m j

5,

any

nonflat Yang-Mills

connection on any ^vector bundle E over the

m-sphere

^{sm with the}

standard

Riemannian metric is un-

stable.

A

proof

of Theorem 1.3 can be found in the paper of

Bourguignon

^{and Lawson}

[B.L].

^{See also}

[K.O.T].

^{On the other}

hand, Laquel [L]

^{showed a} table of the indices and nullities of the canonical connections

(which

^are

typical examples

^of

Yang-

Mills

connections)

^{of all}

compact

irreducible

symmetric

spaces and all

compact

simple

Lie groups. On his

table,

^{the index}

i(~0)

^{of the canonical} connection

Do

on the

m-sphere

^Sm

(m 5)

^with the standard Riemannian metric is

given by

Now our theorem is as follows:

THEOREM 1.5.

(cf. Theorem 4.1)

^For any

nonflat Yang-Mills

connection ~ on

any ^vector bundle E over the

m-sphere

^sm

(m 5)

with the standard Riemannian

metric,

It

might

^{be of some interest to} compare ^our result with

instability

results for harmonic maps.

THEOREM 1.6.

(Xin [X])

^For m ³

and for any

Riemannian

manifold ^N,

^there

is no nonconstant

weakly

^{stable harmonic} maps

f :

^{Sm - N.}

Our

proof

of the above main theorem follows the method of El Soufi

developed

in the

setting

of minimal immersions

(cf. [E1])

and harmonic maps

(cf. [E1], [E2])

on the index

i( f )

^{of a} harmonic map

f :

^{S’ ---+ N:}

THEOREM 1.7.

(El Soufi [E2]) For m ^3, any Riemannian manifold N and any

nonconstant harmonic _map

f :

^{Sm ~}

N,

The estimate

(1.8)

^{is also}

optimal,

ⁱⁿ

fact,

^the

identity

map id: Sm ~ S"2

(m 3)

^{and the}

Hopf fibering

^{x :}

S3 ~ S2

achieve the

equality (cf. [S], [U]).

2. Preliminaries ^- 2.1. YANG-MILLS CONNECTIONS

Let

(M,g)

^{be a}

^compact

Riemannian

manifold,

^{P a}

principal

^{G-bundle over}

M,

^{and E the} associated G-vector bundle of rank r over

M,

^with

projections

03C0: P ~ M and 7r: E ~

M,

^{where G is a}

compact

Lie group. Recall for ^a

given

faithful

representation

^{p: G -}

O(r),

^{E is}

^given

^as

where

[u, y]

^{is an}

equivalence

^class

containing (u, y) E

^{P x}

RT,

^{and the}

equivalence

relation is

(u, y) - (ub, p(b)-ly), b

^{E G. Each} CI section Q E

I(E)

^{can be}

regarded

^{as a CI}

map â

of P into W

by

where 03C0(u)

^{= x} and each u

E P is regarded

^{here as an onto}

isomorphism

u :

RT :3

y -

[u, y]

^E

^Ex,

^where

^Ex

^{is the fiber of E over x E M.}

Via

(2.1),

the connection form w

(cf. [K.N,

p.

64]) corresponds

^{to a}

unique

G-connection V on E

by

where D is the covariant exterior differentiation defined

by

and

Here _x* is the differential of the

projection

^{7r: P - M and}

^WH

^is the horizontal

component

^{of W}

corresponding

^{to the}

decomposition

where

Vu = 1 W

^E

T,, P; 03C0*(W)

⁼

0} ^(the

^vertical

space)

^and

Hu

⁼

{W

^E

TuP; 03C9(W)

⁼

0} ^(the horizontal space).

Let

C(E)

^be the space of all CI G-connections on E. The group of all ^auto-

morphisms

^{of E}

inducing

^the

identity

map of M is called the _{gauge group,} denoted

by Ç( E).

The gauge group

G(E)

is identified with the group ^{of all}

automorphisms

cp of P

satisfying ~(ua)

⁼

~(u)a

^{for u}

^{E P and}

^{a E} G. The identification is

given by G(E) ~ ~ ~ Ç3, where cp

:= ~ ^{o u} with u E P

being

considered as a linear

isomorphism

^{u: RT ~}

Ex.

The group

G(E)

^{acts on}

C(E) by

for Q E

0393(E), ~

^E

g(E)

^{and i7 E}

C(E).

^{We can}

regard

^the space of C°° sections of the vector bundle _{P Xpd g,}

r(P

^xAd

g),

^{as the Lie}

^algebra

^of

9 (E).

^{The bundle}

P Xpd g is identified with a subbundle of

End(E)

^{via p, denoted}

by

^{gE. The}

identification is

given by

Note that

G(E)

^{admits an} affine structure,

i.e.,

the difference of two connections A = B7 -

~’

is in

03A91(gE)

^and

C(E) = {~

⁺

^{A ; A}

^E

03A91(gE)}

^for any fixed B7 E

C(E). Equivalently,

the difference of two connection forms a = w - ce’

is in

03A91(P

^xAd

g)

^{and any} connection form can be

expressed

^{as w + a with}

a E

03A91(P

^xAd

g)

^{for a} fixed connection form w.

To each G-connection i7 on

E,

^the curvature tensor

R7

E

Q2 (OE)

^is

^by

definition

for C°° vector fields X and Y on M. It

corresponds

^{to a}

g-valued

^{2-form n on}

P,

called the

curvature form,

^defined

by

^{n : = dw + 03C9 A 03C9 with w}

being

^{the connection}

form of V. It holds that

equivalently, Q

⁼

*03A9

^with

Q _being

the curvature form of the connection form

*03C9.

Let , &#x3E; be an

Ad(G)-invariant

^inner

product

^{on g,} which induces fiber metrics on r X Ad g and _gE,

respectively,

^denoted

by

^{the same}

symbol

, &#x3E;.

In

general,

^{for a} connection i7 ^{on a} vector bundle F over

M,

^let

d~: 03A9p(F) ~ 03A9p+1(F), ^p

^{0 be the}

corresponding

^exterior differential

operator (cf. [B.L]).

^A

fiber metric , &#x3E; of F induces an inner

product

^on

APT; M

⁰

^Fr together

^with

g, denoted

by

^{the same}

symbol

^, &#x3E;. We denote the associated norm

bey 1111.

^The

global

^inner

product ( , )

^on

QP((F)

is defined

by

for

03C8, ~

^E

S2p(F).

^Let

^6°: 03A9p+1(F) ~ Stp(F), p 0,

be the formal

adjoint

^{of the}

operator dv.

Note that

for ~ ~ G(E)

The

tangent

space ^to the orbit of the gauge group ^at

B7,

considered as a

subspace

of 03A91(gE) ~ TB7C(E),

^is

d~(03A90(gE)).

^Its

orthogonal complement

^{to this}

subspace

in

03A91(gE)

^is

Ker(b7).

^The

^subspace Ker(6V)

^C

03A91(gE)

is called the space of

infinitesimal deformations

of the connection B7.

Now let us recall the

Yang-Mills

functional.

DEFINITION 2.3. The function

yM: C ( E )

^{~ R defined}

by

is called the

Yang-Mills functional.

^{A critical}

point ~

^E

G(E)

^{of the}

^Yang-Mills

functional is called a

Yang-Mills

connection which is

equivalent

^to

bvrv

^{= 0.}

The

Hodge-deRham Laplacian

^{for vector} bundle valued exterior

p-forms

^is

by

definition

Define an

endomorphism 9t v

of the space

03A91(gE) by

for ~

EnI (9E ),

^where

{e1,

^...,

em}

^{is any} local orthonormal frame field on M with

respect

^to g. Then the second variation formula is

given by:

THEOREM 2.4.

(cf. [B.L]) Suppose 17 = ^~0

^{is a}

Yang-Mills

connection and B ⁼

/tlt=o Vt

^E

03A91(gE).

^Then the second variation

of the Yang-Mills functional

is

given by

provided ^03B4~B

⁼

^0,

^where

The

operator S~

is

elliptic

^and

self-adjoint

^{and the} restriction to the space

Ker(6V)

^C

nI(9E)

^has

eigenvalues ^Ai 03BB2

^{··· - oo,} with associated finite dimensional

eigenspaces Ea,, E03BB2,... .

^{Then we can} define the

(Morse)

^index

and the

(Morse) nullity

^{as follows}

(cf. [B.L]).

DEFINITION 2.5. The index of ^a

Yang-Mills

connection V is the dimension

i(V)= dim(~03BB0E03BB).

^The

nullity

of V is the dimension

n(V)

⁼

dim(Eo).

DEFINITION 2.6. A

Yang-Mills

connection V is said to be

weakly

^{stable if}

i(V)

⁼

^{0, i.e.,}

^{the second}

variation

^0.

Otherwise, V

^{is said to be unstable.}

2.2. CONFORMAL TRANSFORMATIONS

Let sm

= {x

^E

Rm; ~x~

⁼

1}

be the unit

sphere

^{and for} any ^{a E}

Rm+1,

^{define a}

C~ vector field â on sm

by

where ( , )

^is the standard

inner product

^on

^Rm+1 and IIxll2 = x, x&#x3E;, x

^E

^Rm+1.

Let It

⁼

03B3at, t ~ R,

^be

a flow

^{of a,}

^i.e., each It

^{is a C°°}

diffeomorphism

^{of SI}

onto itself

satisfying

It is wellknown

(cf. [El], [0])

^that

each It

^{is a} conformal transformation of

Sm,

^Le.

where g

is the standard Riemannian metric on Sm with constant curvature

1,

^and

at is a Coo

positive

^{function on S"2 which is}

given (cf. [E.1]) by

Let D be the Levi-Civita connection

of (Sm, g)

^and

^D

^{that of}

(sm,,:g).

Then

for all C~ vector fields X and Y on 5’"B ^It is known

(cf. [El])

^that

for C°° vector fields X and Y on

Sm,

^where

grad

^{at is the}

gradient

^{vector field of}

at with

respect to g

^{which is}

given (cf. [El]) by

3. Déformations of connections

In what

follows,

^we

always

^assume the base Riemannian manifold

(M, g)

^{is the}

unit

sphere (Sm, g)

^{with the} standard Riemannian

metric 9

^{of constant curvature 1}

and preserve ^the situations in Section 2.

We fix a G-connection i7 on E with a connection form 03C9. For

0%

^{a e}

Rm+1,

let yt be ^a flow of a ^on Sm . We can define ^a

one-parameter family t, t E R

^of

bundle _{maps of E}

lifting

yt ^as the flow of the horizontal lift a* of the vector field à to E. That

is,

^{for each w}

^{E E}

^with

03C0(w)

^{= x, the curve t ~}

t(w)

^{is a horizontal}

lift of the curve t ^~

03B3t(x).

^{We also define a}

one-parameter family ~t, t ~ R,

^of

bundle _{maps of P} so that each u ~ ^P

with r(u) =

^{x, a curve t ~}

~t(u)

^{is the}

parallel transport

^{of u with}

respect

^{to w}

along

^{the curve t ~}

03B3t(x).

Then it holds that

where each u

E P is regarded

^{as an onto linear}

isomorphism

^{u : RT ~}

Ex

^and

yt

^{o u is its}

composition

^with

t: ^{Ex ~} E’Yt(x).

For each CI section a E

0393(E),

^{define a new C°°}

section it . U

^E

0393(E) by

It is known

(cf. [K.N, p. 114])

^that

Moreover,

^{the CI} map of P into RT

corresponding

^{to the}

section t·03C3

^E

0393(E)

as in

(2.1 )

^is

given by

DEFINITION 3.5. For ^a G-connection i7 of E with a connection form w, define

a

one-parameter family

of connections

~t, t ~ R by

for a vector field X on Sm.

LEMMA 3.7.

(i) ~t

^{is a} G-connection with the

connection form ~*t03C9.

(ii)

^Its

infinitesimal deformation satisfies

^that

That

is, for

^a

vector field

^{X on}

S’n,

(iii)

^The

curvature form 03A9~*t03C9

^is

equal

^to

t and for

^CI

vector fields

^X

and Y on

sm,

Proof.

^For

(i),

ⁱⁿ

fact,

^it is easy ^{to see}

~t

is ^a connection ^on E. For

instance,

for

f

^E

C~(Sm),

^{(J E}

0393(E)

^{and a}

^C~

^{vector field X on}

^Sm,

since

d03B3t(X)(f

^o

03B3-1t)

⁼

^{X f} 0 -it

^{and the other} conditions to be ^a connection

are verified in a similar way. On the other

hand,

since ~t ^o

Rb

⁼

Rb

^o pt, where for b _e

G, Rb: P ~ u ~

^{u·b e}

P,~*t03C9 satisfies two

conditions

to be a connection form,

in fact. To see that

~t

is a

G-connection,

^we

only

^{have to see} the connection form of

~t

coincides with

~*t03C9.

^{To see}

^this,

^let

^~t

^be the G-connection

corresponding

to

pgoe. By (2.2),

^{for a E}

0393(E),

Since,

^{for a vector field W on}

P,

and

(ii)

^{For a vector field X on Sm and a E}

r(E),

By (3.3)

^and

1t It=od,tX

^{= -}

^{[a, X],}

^the

right hand

^{side coincides with}

R~(a, X)03C3.

The assertion

(iii)

^{follows from} definitions of

R~t, ^~t

^{and the curvature}

form. 0

LEMMA 3.1 l.

IF i(a)R~

⁼

^0,

^then

Thus if

i(a)R~03C3

^{= 0 for any 03C3 e}

0393(E),

^then

d dt|t=0~*t03C9

^{= 0 because of the}

faithfulness of _p. This

implies ~*t03C9

⁼

03C9, t ~ R.

^D

LEMMA 3.12.

(cf. [B.L,

p.

215]). If ~

^{is a}

Yang-Mills connection,

then B :=

i(a)R~, a

^e

^Rm+1, belongs to the subspace Ker(03B4~) ofnl(gE).

Proof.

This lemma is an immediate consequence of Lemma 7.3 in

[B.L,

p.

215]

since

6V’ RV’

⁼ 0 and a is ^a vector field of

gradient type

^{in the sense of}

[B.L].

⁰

4. Main theorem We shall _prove:

THEOREM 4.1. Let

(Sm, g)(m 5)

^{be the}

m-sphere

with the standard Rieman- nian

metric 9 of constant

^{curvature l. Let} E be any G-vector bundle ^over sm with G a compact Lie group and ~ _any

nonflat Yang-Mills

connection on E. Then

Let

0 ~

^{a e}

^Rm+1

^be

arbitrarily

^{fixed. Let}

~t, t ~ R,

^{be the}

one-parameter family

of G-connections in Definition 3.5. We shall show the

following

^two propo- sitions in the

following

^sections.

PROPOSITION 4.2. Put

Then we obtain

PROPOSITION 4.3.

If i(â)Rv

⁼

0 for some 0 ~

^{a E}

Rm+1,

^{then ~}

_{is flat.}

Proof of

^{Theorem 4.1.}

By

^{these two}

propositions,

^{we obtain}

immediately

Theorem 4.1. In

fact,

^{assume that V is a nonflat}

Yang-Mills

connection on E and m &#x3E; 4. Then

by

Lemma3.l2 and

Proposition 4.3,

^{V =}

{i(a)R~

^E

03A91(gE);

a E

Rm+1}

^{is an}

(m

⁺

1)-dimensional subspace

^of

Ker( 6V),

^and

by Propo- sitions4.2,

the second variation of

y M

^{at B7 is}

negative

^{definite on V.}

Thus,

i(~) m+1.

^~

5. Proof of

Proposition

^4.2

We

only

^{have to}

^show (i)

^since

(ii)

^follows

immediately

^from

(i).

Let

Bt

⁼

d ds|s=0~t+s. ^{By (3.8),}

^{we have}

We first show

where {ej}mj=1

^{is a} local orthonormal frame field on

(Sm,g),

^{and D and}

^D

^are

the Levi-Civita connections of

(Sm, g)

^and

(Sm, 03B3*tg), respectively.

^In

^fact,

^for

(J e

0393(E),

because of the definition of coincides with

Since e’j(03B3tx) :

⁼

03B1t(x)-1/2(d03B3tej)03B3tx, j

⁼

^1,...,

^{m, is a} local orthonormal frame field with

respect to g

and the Levi-Civita connection of

03B3*tg

⁼ atg satisfies

(2.12), (5.3)

coincides with

Notice ^that

since is ^a

Yang-Mills connection).

^{Therefore we}

get

m

which

implies (5.2).

By (2.13),

^we have for any C°° vector field X on

S’n,

3=1

Therefore, substituting (5.4)

^and

(5.5)

^into

(5.2),

Substituting (2.14)

^into

^this,

^{we obtain}

that

is,

Therefore

together

^with

(5.1 ),

^we

finally

^obtain

6. Proof of

Proposition

^4.3.

Let V be a G-connection on E with a connection form w. Assume that

for some

0 ~ a

^E

^Rm+1.

^Then

by Lemma3.11,

where

~t, t ~ R

^{is a}

one-parameter family

^{of C°°} bundle maps of P

corresponding

to a e

Rm+1

defined

by:

^for

^{each u e}

^{P with}

03C0(u)

^{= x, the curve t ~}

’Pt ( u)

is the

parallel transport

^{of V}

along

^{the curve t ~}

03B3t(x) and -it

^{is a flow on sm}

of â.

Then we have

LEMMA 6.2.

(i)

^{We have}

where

Hu

⁼

(X

^E

^TuP; 03C9(X)

⁼

^0}

^{is the}

horizontal subspace of TuP

and ’Pt*

is the

differential of (p

^{at u.}

(6.3)

^{means that} ’Pt sends a

horizontal

^{curve s ~}

u(s)

in P to another

horizontal

^{one s ~}

~t(u(s))

^{in P.}

(ii)

^{Let u E}

PB03C0-1 (

^-

a ~a~).

^{Then u~ =}

^{limt~~ cpt(u)}

exists and lies in

7r (

^-

a ~a~)

^C

^P,

^{and the}

correspondence

^{u ~ u~ is smooth.}

Proof.

The assertion

(i)

^{is an} immediate consequence of

(6.1).

We prove

(ii).

For each x E

SmB{- a ~a~}, limt~~ 03B3t(x)

⁼

a ~a~, and the curve t

^~

03B3t(x) is a

smooth curve with finite

length.

^{Thus we}

get

^{a smooth curve s ~}

c(s) connecting

x and

yâll, reparametrizing by

^{the arc}

length

^{s =}

s(t).

Then for any ^{u E} P with

7r(u)

^{= x,}

’Pt ( u)

^{is the}

parallel transport

^{of u also}

along c(s)

^{with s =}

s(t).

^The

assertion

(ii)

^{is now}

straightforward.

^~

Fix _any

point

^uo

~ 03C0-1(a ~a~). ^{By Lemma6.2,}

^{for each u E}

^{PB03C0-1 (}

^-

Il:11)’

there exists a

unique b(u)

^{E G such that}

Therefore we can define a smooth

mapping $

^of

PB03C0-1 (

^-

a ~a~)

^{into the}

^product bundle (SmB{-a ~a~}) G by

Clearly -*

^{has a} smooth inverse and satisfies

for all u E

PB03C0-1 ( - rfan).

^{Thus 4b}

^gives

^an

isomorphism

between the

princ

G-bundles

PB03C0-1 (

^-

a ~a~)

^and

^{(S- B 1 -} a 1)

^{x G.}

Moreover,

^{we have}

LEMMA 6.6. The

differential ^03A6* of 03A6

^{at u E}

PB03C0-1 (

^-

a ~a~)

^is

^{given by}

where A is a

unique

^element

of

^g such that the vertical

component of X, XV, equals A*u.

Proof.

^{For a} smooth function

f

^{on S"2 x}

G,

^{we shall calculate}

since

b(uc)

⁼

b(u)c

^for

^{c E G} by (6.4)

^and the definition of cpt. Thus ^we

get

03A6*XV = (0, Ab(u»-

As

for 03A6*XH,

take any horizontal curve

1 3

^{s ~}

u(s) E

^{P with}

u( 0)

^{= u and}

u (0)

⁼

X H,

where I is an open interval

containing

^0.

By

^{Lemma 6.2}

(i),

^{for each}

t,

^{the curve s ~}

~t(u(s))

^is

horizontal,

^i.e.

for all s e I.

By Lemma 6.2 (ii), letting t

^{~ oo, we}

get

for

all s E I.

That

is,

^{the curve s -}

u~(s)

^{is also a} horizontal curve. But

Therefore

u~(s), ^{s E I,}

^{is a}

single point.

^Thus

b(u(s))

⁼

b(u)

^{for all s E I. Hence}

we get

We obtain Lemma 6.6. ^o

Due to Lemma 6.6, for all X

⁼

XH+XV where XH

E

Hu and XV = A*u ^{E Vu}

with A e g, ^we

get

This means

that 03A6-1*03C9

coincides with the canonical flat connection form on the

product

^bundle

(SmB{- a 1)

^{X G}

^{(cf. [K.N, p. 92]).}

Therefore the connection form w

corresponding

^{to V}

gives

^a flat connection on the G-bundle

PB03C0-1 ( -a ~a~)

(cf. [K.N,

p.

92]).

^Since

PBir-1 (

^-

a ~a~)

^{is an open} and dense subset of P and the curvature form QW is

continuous, w

^{is a flat} connection on P. We obtain

Proposition 4.3.

⁰

REMARK 6.7. In the ^{case m =}

4,

for any

Yang-Mills

connection V _{of any} G-vector bundle ^over

Sm,

^the

nullity

is estimated ^as

n(~) m + 1 = 5,

by

^{a similar}

argument.

References

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